3 Answers
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The scene is the culmination of the movie-long reveal of the origins of the killer creatures that cause so much havoc in "Alien" and its sequels.

In "Prometheus," the creature is the result of a somewhat complicated process:

A human (Holloway) was infected by the black goo

The infected human created a new organism within a non-infected human (Shaw)

This new organism grew to be an octopus-like mutant

The octopus-like mutant created another new organism within an Engineer

The second new organism's "birth" consisted of it springing out of its host's rib cage, just like the creatures in "Alien" and its sequels -- which it also resembles physically. (Notably, the alien at the end of Prometheus looks somewhat different from the aliens in the other Alien movies. I attribute this to the fact that in Prometheus, the alien is spawned from an Engineer, while in the other movies the aliens are spawned from humans.)

One of movie's ironies is the first action on the list above was taken by an android, who was a creation of man, who in turn was a creation of the Engineers.

In my opinion the movie hadn't too many argument... I was waiting for something better, but this came to discussion with a friend after we saw the movie and we had different theories. But yes, it's complicated, and the story could have lead to something much better.
– NuxyAug 14 '12 at 6:16

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the problem about this is that in the predator franchise, aliens are already present in the whole universe for along time and considered an own species
– oersAug 18 '12 at 9:14

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Also please don't forget that some of Engineer bodies that were found by geologist and biologist shortly before their death had their rib cages burst from inside - typical for alien birth.
– binarycodeSep 13 '12 at 20:49

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@oers and binarcode: the answer doesn't say that the Prometheus scene showed the FIRST alien birth -- just AN alien birth. To make sure the audience doesn't miss that this is how aliens get made.
– Shiz Z.Sep 13 '12 at 21:32

Everything Shane said is correct, but I believe the true meaning of the birth is an answer to one of the biggest questions of the movie; what caused the chaos on LV-223 that occurred 2000 years prior to the Prometheus expedition? We know it had something to do with the black goo because of how the engineer's decapitated-head reacted when its nervous system was revived by Ford (The head shows symptoms similar to the ones experienced by Holloway after being infected with the goo).
Well remember when David first activates the engineer holo-record-log-thingy? If you listen closely during the opening moments of the log, you can hear a faint screech right before the engineers appear fleeing from an unknown threat. I noticed that the screech is extremely similar to the screech of the new-born xenomorph at the end of the film. So it is possible that the problem that led to the disruption of the engineer's plan to exterminate humanity was caused by an engineer-born xenomorph. If my thesis is correct then the writers of Prometheus outdid themselves in a very odd way.

I agree with Shane; and the birth of an alien from the engineer is to merely show a greater association of Prometheus to the original Alien franchise...the fact that the alien looked different is because of who it was hatched from. Now we know that the Black Goo in the movie is associated with the creation of the Alien creatures.